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Regarding the Blood of Jesus itself

In order to answer the question, we need to define what exactly is meant by “blood.”

If we mean the physical bodily fluid that coursed through the veins and arteries of Jesus, then clearly it is an aspect of Jesus’ human nature, hence human and finite.

However, “blood” can be a symbolic reference to the Passion and Death of Jesus. In that case, although Jesus felt the physical effects of the Passion in his human nature, it was still his Divine Person who underwent it. The capacity of the Passion to redeem us, therefore, is infinite. (Regarding the Passion, see the Catechism of the Catholic Church [CCC], 603-623).

Finally, “blood” can refer to Eucharistic species of the Precious Blood. Although the Precious Blood has the appearance of wine (and thus in that respect resembles Jesus’ physical blood), in reality Jesus in his totality is present in the species of wine; in other words, it is not merely his physical blood. In that sense, the Precious Blood includes both natures: the human (and thus finite) and Divine (and thus infinite). (See CCC 1377.)

Regarding the redemptive value of Jesus’ actions

The broader question regards something called the Hypostatic Union: although the Divine Son remains the Divine Son (indeed, remains unchanged), by the Incarnation, He took on a (finite) human nature. Consequently, the Son was able to do all of the things that a man can do: be born, eat, sleep, laugh, and even die.

Nevertheless, the Person who was born, ate, slept, laughed, and even died, was none other than God Himself. (See CCC 466.)

The consequence of this fact is that all of Jesus’ actions and passions—even those accomplished in his finite human nature—have an infinite value, precisely because a Divine Person is doing them.

That is how Jesus’ finite human nature could be the instrument of something (the Passion) that was of infinite redemptive value.

Thanks AthanasiusOfAlex. So his blood is both finite and infinite, temporal and eternal, correct?
– anonymouswhoMay 6 '17 at 6:46

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@anonymouswho In those different respects, yes. His physical blood (the bodily fluid) is finite and temporal, as are the physical effects of his Passion. However, they both take on an infinite value because of their union with the Divine Person of the Son.
– AthanasiusOfAlexMay 6 '17 at 6:53

I think I can see how the human sacrifice has an "infinite value" in that the sacrifice is a concept of continuous action, rather than a substance. What is an "infinite value" when ascribed to a substance such as blood? Does his blood exist as an actual infinite (while also being finite, of course)?
– anonymouswhoMay 6 '17 at 7:21

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@anonymouswho The infinity in question is Divine Infinity (if we can call it that), which is a kind of actual infinity. However, we should not confuse that with an actual infinity of extension (either in time or in space). It similar to what we mean when we say that God is omniscient, omnipresent, omnipotent, etc. Thus, it is neither the blood nor the action, in itself, that has infinite value, but rather the union of each to the Divine Person. (By “value,” here, I mean chiefly a thing’s capacity to have redemptive effects on man.)
– AthanasiusOfAlexMay 6 '17 at 7:30